- #1
nonequilibrium
- 1,439
- 2
Hello,
After a theorem stating that the product, sum, etc of two elements of a field extension that are algebraic over the original field are also algebraic, my course states the following result (translated into english):
but later in my course it defines "the algebraic closure of F" as a field extension of F that is
(i) algebraically closed (in the sense that every polynomial has a root)
(ii) algebraic across F
These seem to be different concepts, am I right? Because the former doesn't need to be algebraically closed (despite its name...), because for example "the algebraic closure of [itex]\mathbb Q[/itex] in [itex]\mathbb R[/itex]" still has no solution for X²+1=0, yet "the algebraic closure of [itex]\mathbb Q[/itex]" (full stop) does.
So is the only difference seperating these two concepts the suffix/appendix "in E"?
After a theorem stating that the product, sum, etc of two elements of a field extension that are algebraic over the original field are also algebraic, my course states the following result (translated into english):
[itex]\textrm{Let $F \subset E$ be fields. The elements of $E$ that are algebraic across F form a subfield of $E$ (and of course a field extension of $F$).}[/itex]
[itex]\textrm{We call this subfield the algebraic closure of $F$ in $E$.}[/itex]
but later in my course it defines "the algebraic closure of F" as a field extension of F that is
(i) algebraically closed (in the sense that every polynomial has a root)
(ii) algebraic across F
These seem to be different concepts, am I right? Because the former doesn't need to be algebraically closed (despite its name...), because for example "the algebraic closure of [itex]\mathbb Q[/itex] in [itex]\mathbb R[/itex]" still has no solution for X²+1=0, yet "the algebraic closure of [itex]\mathbb Q[/itex]" (full stop) does.
So is the only difference seperating these two concepts the suffix/appendix "in E"?